Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110517352571 -0.82 0.80 8.0 2.92s
2 -1.117201917117 -2.17 -1.86 0.80 1.0 1.36s
3 -1.117250500976 -4.31 -2.70 0.80 1.0 19.3ms
4 -1.117251172809 -6.17 -3.58 0.80 1.0 19.0ms
5 -1.117251188386 -7.81 -4.00 0.80 1.0 19.3ms
6 -1.117251189476 -8.96 -4.22 0.80 1.0 53.2ms
7 -1.117251190120 -9.19 -5.51 0.80 1.0 20.8ms
8 -1.117251190143 -10.66 -5.88 0.80 2.0 23.0ms
9 -1.117251190143 -12.29 -6.30 0.80 1.0 20.7ms
10 -1.117251190143 -13.51 -6.46 0.80 1.0 28.8ms
11 -1.117251190143 -13.62 -6.99 0.80 1.0 21.3ms
12 -1.117251190143 -14.15 -7.49 0.80 1.0 21.2ms
13 -1.117251190143 + -14.40 -8.12 0.80 1.0 192ms
14 -1.117251190143 -14.65 -9.09 0.80 1.0 22.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.53 0.80 1.0 3.12s
2 -1.117251190143 -14.88 -10.13 0.80 1.0 21.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 18.9s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997991 -2.46 0.80 1.0 13.7ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 59.3ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 19.3ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 19.5ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.6ms
6 -1.117520613465 -12.14 -6.42 0.80 1.0 35.8ms
7 -1.117520613465 -13.35 -7.40 0.80 1.0 21.1ms
8 -1.117520613465 -14.57 -8.21 0.80 1.0 22.0ms
9 -1.117520613465 -15.05 -8.94 0.80 1.0 24.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 13.2ms
2 -1.118339043932 -4.04 -2.70 0.80 1.0 25.1ms
3 -1.118340011111 -6.01 -3.44 0.80 1.0 19.1ms
4 -1.118340028552 -7.76 -4.36 0.80 1.0 19.0ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 25.2ms
6 -1.118340029163 -10.57 -5.62 0.80 1.0 20.0ms
7 -1.118340029165 -11.73 -6.68 0.80 1.0 22.6ms
8 -1.118340029165 -13.53 -7.58 0.80 1.0 20.9ms
9 -1.118340029165 + -15.05 -8.12 0.80 1.0 23.2ms
10 -1.118340029165 -14.88 -8.55 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -9.05 0.80 1.0 13.3ms
2 -1.118340029165 + -14.95 -10.36 0.80 1.0 115ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029165 -11.04 0.80 1.0 518ms
2 -1.118340029165 + -14.29 -11.42 0.80 1.0 19.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029165 │ │ 0.00297055 │ 3.62s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 14.3ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 23.3ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 25.1ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 26.9ms
5 -1.118346866261 -12.09 -6.55 0.80 1.0 27.2ms
6 -1.118346866261 -13.44 -7.04 0.80 1.0 25.3ms
7 -1.118346866261 -14.61 -8.13 0.80 1.0 24.3ms
8 -1.118346866261 -14.54 -8.95 0.80 1.0 23.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 14.6ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 20.3ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 20.1ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 23.3ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 27.5ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 28.2ms
7 -1.118356101386 -13.63 -7.65 0.80 1.0 36.1ms
8 -1.118356101386 + -15.35 -8.45 0.80 1.0 20.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.2ms
2 -1.118356101386 + -14.45 -9.82 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.6ms
2 -1.118356101386 + -15.35 -11.46 0.80 1.0 18.6ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 588ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 17.7ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.8ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.9ms
4 -1.118356102877 -13.97 -7.62 0.80 1.0 19.3ms
5 -1.118356102877 + -14.45 -8.39 0.80 1.0 19.6ms
6 -1.118356102877 + -Inf -8.89 0.80 1.0 20.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.8ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 19.0ms
4 -1.118356105120 -13.21 -7.09 0.80 1.0 19.4ms
5 -1.118356105120 + -Inf -7.86 0.80 1.0 22.6ms
6 -1.118356105120 -14.48 -8.35 0.80 1.0 20.2ms
7 -1.118356105120 + -15.35 -9.45 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 16.4ms
2 -1.118356105120 -14.29 -11.02 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 13.3ms
2 -1.118356105120 + -14.48 -12.44 0.80 1.0 21.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.01329e-8 │ 481ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -7.34033e-18, 6.25818e-18]u"a₀")
Atom(H, [ 1.44318, 1.62145e-17, -2.85384e-17]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658680663034 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335118717925 -0.80 0.80 8.0 797ms
2 -8.340932815481 -2.24 -1.69 0.80 1.3 619ms
3 -8.341205451924 -3.56 -2.77 0.80 2.7 39.0ms
4 -8.341214229026 -5.06 -3.29 0.80 3.3 41.7ms
5 -8.341214267975 -7.41 -4.02 0.80 1.3 28.7ms
6 -8.341214272350 -8.36 -5.04 0.80 2.0 51.0ms
7 -8.341214272510 -9.80 -5.39 0.80 3.7 42.5ms
8 -8.341214272518 -11.12 -6.15 0.80 2.3 33.8ms
9 -8.341214272518 -12.44 -7.55 0.80 3.0 38.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.46 0.80 1.0 612ms
2 -8.341214272518 -14.15 -9.38 0.80 1.0 39.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.64 0.80 1.0 23.3ms
2 -8.341214272518 + -14.75 -9.75 0.80 1.0 26.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.0842e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350381304631 -1.20 0.80 9.7 70.6ms
2 -8.351862846972 -2.83 -1.88 0.80 2.0 30.3ms
3 -8.351927832577 -4.19 -2.78 0.80 3.7 39.1ms
4 -8.351928491701 -6.18 -3.76 0.80 3.7 40.1ms
5 -8.351928514481 -7.64 -4.92 0.80 3.3 73.8ms
6 -8.351928514631 -9.82 -5.66 0.80 4.0 43.5ms
7 -8.351928514635 -11.44 -6.78 0.80 3.3 39.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353363879313 -1.52 0.80 8.7 66.1ms
2 -8.353724286524 -3.44 -2.21 0.80 1.0 25.3ms
3 -8.353757341487 -4.48 -2.98 0.80 4.3 42.8ms
4 -8.353757520586 -6.75 -4.22 0.80 2.3 31.6ms
5 -8.353757531683 -7.95 -4.99 0.80 3.7 41.3ms
6 -8.353757531732 -10.32 -5.70 0.80 2.7 52.2ms
7 -8.353757531734 -11.68 -7.17 0.80 2.3 34.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531734 -8.05 0.80 1.0 22.7ms
2 -8.353757531734 -14.45 -8.29 0.80 1.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531734 -8.85 0.80 1.0 22.9ms
2 -8.353757531734 + -Inf -9.01 0.80 1.0 25.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531734 -9.48 0.80 1.0 23.2ms
2 -8.353757531734 -14.27 -9.57 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531734 │ │ 1.00266e-14 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353693542993 -3.14 0.80 8.0 71.1ms
2 -8.353749397313 -4.25 -3.79 0.80 1.0 25.6ms
3 -8.353770531477 -4.68 -3.74 0.80 6.0 55.0ms
4 -8.353770536608 -8.29 -4.08 0.80 1.0 32.6ms
5 -8.353770536655 -10.32 -4.29 0.80 2.0 30.7ms
6 -8.353770537346 -9.16 -4.91 0.80 1.7 28.3ms
7 -8.353770537405 -10.23 -5.47 0.80 3.0 35.3ms
8 -8.353770537410 -11.29 -6.52 0.80 2.7 38.5ms
9 -8.353770537410 -12.89 -6.91 0.80 3.3 41.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353760713935 -2.00 0.80 9.3 75.3ms
2 -8.353881791214 -3.92 -2.75 0.80 1.0 25.4ms
3 -8.353895070193 -4.88 -3.44 0.80 5.0 50.2ms
4 -8.353895107262 -7.43 -4.14 0.80 3.0 37.0ms
5 -8.353895110697 -8.46 -5.37 0.80 2.7 35.3ms
6 -8.353895110765 -10.17 -5.97 0.80 3.7 42.7ms
7 -8.353895110765 -13.15 -6.88 0.80 1.7 33.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -7.62 0.80 1.0 23.7ms
2 -8.353895110765 + -14.45 -8.52 0.80 1.0 30.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -9.00 0.80 1.0 23.0ms
2 -8.353895110765 + -Inf -9.26 0.80 1.0 25.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110765 -9.58 0.80 1.0 23.1ms
2 -8.353895110765 -14.27 -9.92 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110765 │ -3.86 │ 6.98533e-15 │ 0.000646187 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353822775824 -3.56 0.80 9.0 68.5ms
2 -8.353884839874 -4.21 -3.80 0.80 1.3 26.8ms
3 -8.353895211803 -4.98 -3.95 0.80 6.3 62.6ms
4 -8.353895212004 -9.70 -4.14 0.80 1.0 25.8ms
5 -8.353895212465 -9.34 -4.57 0.80 1.3 27.7ms
6 -8.353895212500 -10.45 -4.82 0.80 1.3 30.7ms
7 -8.353895212512 -10.94 -5.40 0.80 1.7 29.5ms
8 -8.353895212513 -11.87 -5.96 0.80 2.0 31.5ms
9 -8.353895212513 -12.90 -6.61 0.80 2.3 33.8ms
10 -8.353895212513 -14.05 -6.58 0.80 2.7 39.7ms
11 -8.353895212513 -13.97 -7.53 0.80 1.0 26.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353832797751 -3.34 0.80 8.7 72.3ms
2 -8.353884595356 -4.29 -3.94 0.80 1.0 25.7ms
3 -8.353895361592 -4.97 -4.02 0.80 6.3 57.2ms
4 -8.353895362591 -9.00 -4.50 0.80 1.3 31.4ms
5 -8.353895362650 -10.23 -4.86 0.80 2.0 31.3ms
6 -8.353895362683 -10.49 -5.51 0.80 2.0 30.9ms
7 -8.353895362685 -11.58 -5.83 0.80 2.7 38.1ms
8 -8.353895362686 -12.37 -7.33 0.80 1.7 30.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.66 0.80 1.0 22.7ms
2 -8.353895362686 -14.27 -8.07 0.80 1.0 29.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.54 0.80 1.0 22.8ms
2 -8.353895362686 -14.75 -8.77 0.80 1.0 26.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.10 0.80 1.0 22.5ms
2 -8.353895362686 -14.75 -9.32 0.80 1.0 25.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.72506e-15 │ 1.71679e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-4.60198e-18 5.27783 5.27783;
5.27783 -3.22012e-18 5.27783;
5.27783 5.27783 -7.36645e-18]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662917851517 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.